13 LIMITS 49

13. LIMITS

Let E be a metric space, A a subset of E, a a cluster point of A. Suppose
first that a does not belong to A, Then, if/is a mapping of A into a metric
space E', we say that f(x) has a limit af e E' when x e A tends to a (or also
that a' is a limit of fat the point aeA with respect to A), if the mapping g of
A u {a} into E' defined by taking g(x) =f(x) for x E A, g(a) = a', is con-
tinuous at the point a\ we then write a' = lim /(X). If a e A, we use the

*-+«, #eA

same language and notation to mean that/is continuous at the point a, with
a' =f(a).

(3.13.1) In order that a' e E' be limit off(x) when x E A. tends to a, a necessary
and sufficient condition is that, for every neighborhood V of ar in E', there
exist a neighborhood Vofa in E such thatf(V n A) e V.

(3.13.2) In order that a1 e E' be limit off(x) when x e A tends to a, a necessary
and sufficient condition is that, for every e > 0, there exist a 6 > 0 such that
the relations x e A, d(x, a) < 5 imply d'(a',f(x)) < e.

These criteria are mere translations of the definitions.

(3.13.3) A mapping can only have one limit with respect to A at a given
point aeA.

For if a', b' were two limits of/at the point a, it follows from (3.13.2)
and the triangle inequality that, for any e > 0, we would have d\a\ b') ^ 2s,
which is absurd if a' ^ b'.

(3.13.4) Let f be a mapping ofE into E'. In order that f be continuous at a
point x0eE such that x0 is a cluster point ofE — {x0} (which means *0 is not
isolated in E (3.10.10)), a necessary and sufficient condition is that
/(*„) = lim f(x).

Mere restatement of definitions.he topology of E2).